Method moments

Let us consider now two small volumes vq centered at fq and T2, respectively. The number of i-kind particles therein, Ni r ) and iVj(r2), are stochastic quantities which averaged over uq are Ni f ) — Ni f2) = rii t)vo = iV). Define now the correlation functions of similar particles [36, 37] 

Here the average is taken over a whole volume with a fixed f = f — f2 distance. Due to the system s homogeneity and isotropy the Si and Sij are functions of r = f only. We illustrate below how the equations for the correlation functions could be derived from the set of equations (2.1.40) and the averaging procedure. First, let us define the initial conditions for S i(r,0) and Sij r,Q). 

For non-overlapping volumes, r Aq, the fluctuations in particle numbers are statistically independent and thus 

Taking into account the statistical independence in coordinates of dissimilar particles, we arrive at 

Neglecting third-order momenta, the joint correlation functions (2.1.85), (2.1.86) become [36, 37] 

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The various studies attempting to increase our understanding of turbulent flows comprise five classes moment methods disregarding probabiUty density functions, approximation of probabiUty density functions using moments, calculation of evolution of probabiUty density functions, perturbation methods beginning with known stmctures, and methods identifying coherent stmctures. For a thorough review of turbulent diffusion flames see References 41—48. 

Figure 2.5-1 illustrates the fact that probabilities are not precisely known but may be represented by a "bell-like" distribution the amplitude of which expresses the degree of belief. The probability that a system will fail is calculated by combining component probabilities as unions (addition) and intersection (multiplication) according to the system logic. Instead of point values for these probabilities, distributions are used which results in a distributed probabilitv of system fadure. This section discusses several methods for combining distributions, namely 1) con olution, 2i moments method, 3) Taylor s series, 4) Monte Carlo, and 5) discrete probability distributions (DPD). 

The moments method does not attempt to calculate the system s uncertainty distribution directly but calculates the moments of the distribution from which the distribution may be found by an expansion in its moments. 

Figure 16.4 Dlustration of the Fast Multiple Moment method...

Vibrational Spectrum of Quartic Oscillator by Moments Method... 

The quantities AUMC and AUSC can be regarded as the first and second statistical moments of the plasma concentration curve. These two moments have an equivalent in descriptive statistics, where they define the mean and variance, respectively, in the case of a stochastic distribution of frequencies (Section 3.2). From the above considerations it appears that the statistical moment method strongly depends on numerical integration of the plasma concentration curve Cp(r) and its product with t and (r-MRT). Multiplication by t and (r-MRT) tends to amplify the errors in the plasma concentration Cp(r) at larger values of t. As a consequence, the estimation of the statistical moments critically depends on the precision of the measurement process that is used in the determination of the plasma concentration values. This contrasts with compartmental analysis, where the parameters of the model are estimated by means of least squares regression. 

Identical results were obtained using the kinetic theory moment method by Kikuchi, Pooley, Ryder, and Yeomans [28, 29]. 

A moment method to estimate linear mass transfer coefficients for noneliminating organs was derived by Gallo et al. [52], hi is estimated as... 

JM Gallo, FC Lam, DG Perrier. Moment method for the estimation of mass transfer coefficients for physiological pharmacokinetic models. Biopharm Drug Dispos 12 127-137, 1991. 

Diazaphosphorinanes exist as a mixture of three conformers in solution [Eq. (51)], although all heteroatoms in the ring have substituents. Conformational equilibrium is due to the low inversion barrier of nitrogen studied by H NMR and the dipole moment method (83MI1 84MI1) [Eq. (51)]. The results are presented in Table III. 

As discussed in Chapter 2, a fully developed turbulent flow field contains flow structures with length scales much smaller than the grid cells used in most CFD codes (Daly and Harlow 1970).29 Thus, CFD models based on moment methods do not contain the information needed to predict x, t). Indeed, only the direct numerical simulation (DNS) of (1.27)-(1.29) uses a fine enough grid to resolve completely all flow structures, and thereby avoids the need to predict x, t). In the CFD literature, the small-scale structures that control the chemical source term are called sub-grid-scale (SGS) fields, as illustrated in Fig. 1.7. 

In Chapter 5, we will review models referred to as moment methods, which attempt to close the chemical source term by expressing the unclosed higher-order moments in terms of lower-order moments. However, in general, such models are of limited applicability. On the other hand, transported PDF methods (discussed in Chapter 6) treat the chemical source term exactly. 

For a one-step reaction, most commercial CFD codes provide a simple fix for the fast-chemistry limit. This (first-order moment) method consists of simply slowing down the reaction rate whenever it is faster than the micromixing rate ... 

Harrington, R.F., 1968, Field computation by Moment Methods (Krieger, Malabar, Fla.). 

However, the moment method could again be used to give differences in the moments. This information could then be used with Eqs. (36) and (37) to find the dispersion coefficient. 

Leonard, E. F., and Ruszkay, R. J., Frequency, transient and moment methods in process analysis. Paper presented at A.I.ChiE. Meeting, New York, December, 1961. (I)... 

In the moment method, the pseudopotential is expanded in a power series, the first term of which is equivalent to the one-fluid theory... 

Fickett reports that the first order result of the moment method (one-fluid theory) is a rigorous upper bound to the Gibbs free energy, and that the pseudo-pair-potential result is a rigorous lower bound to the same quantity. Both bounds are so widely separated that they are mostly of theoretical interest. Fickett concludes that none of the pseudopotential results is simple enough to use in the complete detonation calculation... 

Fig. 7. Analysis of Schlieren curves for the transport of PVP 360 (initially at 5 kg m-3 to zero concentration gradient across the boundary) in dextran T10 solution. The transport coefficient T was obtained by diffusional analysis using the reduced height-area ratio method ( ), the reduced second-moment method (A) and the width at half-height method ( )47)...

Analytical solutions of quantum Fokker-Planck equations such as Eq. (63) are known only in special cases. Thus, some special methods have been developed to obtain approximate solutions. One of them is the statistical moment method, based on the fact that the equation for the probability density generates an infinite hierarchic set of equations for the statistical moments and vice versa. 

The conformational characteristics of dimethyl esters of dicarboxylic acids are studies by the NMR and dipole moment method. Conformational energies of the internal CH2-CH2 bonds are determined from the observed 1H-1H vicinal coupling constants. Preferred conformations around the C-C bond are elucidated from the RIS analysis of dipble moments. With the RIS parameters thus established, the orientational correlation between the terminal ester groups is examined. The analysis provides the reason why the odd-even effect in the dipole moment is moderate, and attenuates rapidly with n in the ct.co-diester series. 

One of the present authors has extensively used the dipole-moment method to calculate conformational equilibria of saturated heterocycles. In hindsight this has been a frustrating experience not so much because of the assumptions and approximations that must be made, but because the results in some cases are in good agreement with those derived from other methods, whereas for other groups of compounds the dipole-moment conclusions are clearly incorrect. In this discussion we first discuss the method, using piperidines as an example, and then attempt to assess its areas of applicability and causes of failure. 

However, in many other series results have been obtained that are compatible with those from other methods and that gave the dipole-moment method an appearance of general reliability now known to be unjustified. Such compatible results include spiropiperidines (Section III,A,4), tropanes (Section III,B,4), 2-alkyltetrahydro-l,2-oxazines (Section III,C,2), perhydro-pyrido[l,2-c][l,3]oxazines (Section III,D,IX perhydropyrido[l,2-c][l,3]thi-azines (Section III,D,2), dialkylhexahydropyrimidines and perhydropyrido-[l,2-c]pyrimidines (Section III,D,3), 5-alkyldihydro-l,3,5-dithiazines (Section III,G,3), 3,5-dialkyltetrahydro-l,3,5-thiadiazines (Section III,G,4) and, in part, l,2,4,5-tetraalkylhexahydro-l,2,4,5-tetrazines (Section III,H,4) as well as piperidines, tetrahydro-l,3-oxazines, and tetrahydro-l,3-thiazines containing an N-H group. 

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